Topological stability in set-valued dynamics

Roger Metzger; Carlos Arnoldo Morales Rojas; Phillipe Thieullen

doi:10.3934/dcdsb.2017115

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2017, Volume 22, Issue 5: 1965-1975. Doi: 10.3934/dcdsb.2017115

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Topological stability in set-valued dynamics

1.
Instituto de Matemàtica y Ciencias Afines (IMCA), Universidad Nacional de Ingeniera Calle Los Biòlogos 245, 15012 Lima, Perù
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530,21945-970 Rio de Janeiro, Brazil
3.
Institut de Mathématiques Université de Bordeaux Ⅰ, 33405, Talence, France

* Corresponding author: Carlos Arnoldo Morales Rojas
* Corresponding author: Carlos Arnoldo Morales Rojas

Received: November 2016

Revised: January 2017

Published: August 2017

Work partially supported by CNPq from Brazil and MATHAMSUD 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.

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Abstract

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

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Mathematics Subject Classification: Primary:54H20;Secondary:54C60.

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References

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